Corollary 2.3.11. Let \((Y,A)\) be an NDR-pair with \(A\) nonempty, and assume that \(A\) and \(Y\) have the homotopy type of cell complexes. Then the quotient map \(Y \to Y/A\) induces an isomorphism \[ \Pi _{\infty }(Y) \sqcup _{\Pi _{\infty }(A)} * \cong \Pi _{\infty }(Y/A) \] in \(\An _*\). In particular, this applies whenever the inclusion \(A\hookrightarrow Y\) is collared.
Proof. The NDR condition implies that the canonical map from the homotopy cofiber of \(A\hookrightarrow Y\) to the actual quotient \(Y/A\) is a homotopy equivalence; see May (1999), Chapter 8, Section 4. The claim now follows by applying Proposition 2.3.10 to the diagram \(Y \leftarrow A \to *\). □
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